LRFD
Critical Section for Shear
Due to the structural system configuration and the orientation of loads on precast girder bridges,
compression is introduced into the end regions of the girders due to applied loads. For such cases,

LRFD Art. 5.8.3.2/5.7.3.2 states that the critical section for shear shall be taken as the larger of
0.5dvcotθ or dv from the internal face of the support. Since dv is a fixed value and since it is
conservative to adopt a section closer to the support, the critical section for shear in Eriksson Girder is
assumed to be dv from the face of the support.
LRFD Art. 5.8.2.9/5.7.2.8 defines the parameter dv as the effective shear depth, which is taken as
the distance between the resultants of the compressive and tensile forces. In practical terms, dv
is the distance between the centroid of the compression block and the centroid of the tensile
force, taking into account the effects of both prestressed strand and non-prestressed reinforcement
(i.e., rebar).
However, dv need not be taken to be less than the greater of 0.9de and 0.72h, where de is defined
above (see Section 5.8) and h is the total section height.

Using the above rules and assumptions, Eriksson Girder conservatively computes the location of the critical
section for shear when the LRFD Specification is selected as 0.72h from the face of the support or,
measured from the centerline of bearing, as:
Critical Section for Shear = 0.72h + (width of bearing pad)/2 Modified Compression Field Theory
Procedure
One of the biggest changes introduced in the LRFD Specifications is the procedure used to compute the concrete contribution to the
vertical shear strength of the beam. While the basic steps to designing shear steel are the same as for the Standard Specifications,
the procedure for determining Vc has completely changed. Vc must be determined using the Modified Compression Field Theory Method, or
the Sectional Design Model (Art. 5.8.3/5.7.3), as it is termed in the LRFD Specs. In 2008 the calculation of θ and β
was modified to include the calculation of εs instead of εx. Theta, and beta can now be calculated
directly, without the iterations the previous method required.
The new parameter, εs, is calculated using:

⎡| Mu | + 0.5N + | V − V | − A f ⎤

⎢ d u u p ps po ⎥

ε = ⎣ v ⎦

(LRFD Eq. 5.8.3.4.2-3/5.7.3.4.2-2)

s As + Ep Aps

Note that there is a lower limit on Mu equal to |Vu-Vp|dv.

The parameters θ and β can then be calculated as per the following:

4.8
(LRFD Eq. 5.8.3.4.2-1/5.7.3.4.2-1)
(1 + 750ε s )

θ = 29 + 3500ε s

(LRFD Eq. 5.8.3.4.2-3/5.7.3.4.2-3)

Vc is then calculated as per Step 7 below.
The iterative procedure was moved to Appendix B-5 in 2008. The basic steps of calculating εx, θ
and
β are outlined below.
Step 1: Compute the factored shear, Vu

Step 2: Compute the shear stress at the section using:

Eriksson Girder 4 User Manual 146

Eriksson Girder 4 User Manual 147
Vu − ϕVp
ϕbv dv
v =
(LRFD Eq. 5.8.2.9-1/5.7.2.8-1)

Step 3: Assume a value of theta, the crack angle.
Step 4: Compute the strain in the longitudinal reinforcement using:
0.5Nu

• 0.5(Vu
  − Vp
  ) co θ − Aps
  f po

2(E A + E A + E A )s s p ps
v
u
x
d
M
+

⎥
⎦
⎡
⎢
⎤

⎣

ε (LRFD Eq. B5.2-1)

Step 5: Using Table B5.2-1, look up a revised value of theta. Eriksson Girder uses a double interpolation
procedure to determine theta from the values given in the table.

Step 6: If the new value of theta differs from the assumed value by more than 0.1 degrees, Eriksson Girder continues
to iterate by recomputing εx and looking up a new value of theta.

Step 7: Once the value of theta has converged, beta can be determined from Table B5.2-1. Once beta
is known, Vc can be computed using:
d
Vc = 0.0316 β
f ' b
(LRFD Eq. 5.8.3.3-3/5.7.3.3-3)

Note: A question that arises frequently regarding Eq. 5.8.3.3-3/5.7.3.3-3 is the significance of
the coefficient 0.0316. This value represents a conversion factor that is necessary since the value
of f’c is in units of ksi. It is computed as follows:
0.0316
1000
1
≅
Now that Vc is known, the required area of shear steel can be computed in a manner similar to that
of the Standard Specifications.
Standard Specifications
Under the Standard Specifications, the familiar Vci/Vcw method is typically used to determine Vc.
However, for continuous structures, the option is given to compute Vc using the following
simplified equation for the end quarter of interior girders and the pier end of exterior girders:
Vc = 2
f ' b' d
Eriksson Girder computes Vci in accordance with the equations provided in the Standard Specifications, but
with the following modification. The formula for computing Vci is given below:
max
iMcr
= 0.6 '

ci c
M
V b' d + V

• V
  (Std Eq. 9-27)

It has been suggested that the ratio Mcr/Mmax in the above equation be limited to unity (Puckett &
Tadros, 2001). However, since this restriction is not mentioned in the Standard Specifications at
this time, but it is recognized that it is reasonable to establish an upper bound, this ratio is
limited to 100 by Eriksson Girder. Note that when this ratio becomes large, Vci becomes very large and Vcw
will govern. The calculation of Vc is, therefore, self-limiting, which may be what the
specifications had intended.